DES and RES Processes and their Explicit Solutions

Transcription

1 DES and RES Processes and their Explicit Solutions Michael N Katehakis Dept of Management Science and Information Systems, Rutgers Business School - Newark and New Brunswick, 1 Washington Park Newark, NJ 07102, mnk@rutgersedu Laurens Smit Mathematisch Instituut, Universiteit Leiden Niels Bohrweg 1, 2333 CA, The Netherlands, & Dept of Management Science and Information Systems, Rutgers Business School - Newark and New Brunswick, 1 Washington Park Newark, NJ 07102, laurenssmit@rutgersedu Flora Spieksma Mathematisch Instituut, Universiteit Leiden Niels Bohrweg 1, 2333 CA, The Netherlands, spieksma@mathleidenunivnl This article defines and studies the down entrance state (DES) and the restart entrance state (RES) classes of quasi skip free (QSF) processes specified in terms of the nonzero structure of the elements of their transition rate matrix Q A QSF process is a Markov chain with states that can be specified by tuples of the form (m, i), where m Z represents the current level of the state and i Z + the current phase of the state, and its transition probability matrix Q does not permit one step transitions to states that are two or more levels away from the current state in one direction of the level variable m A QSF process is a DES process if and only if one step down transitions from a level m can only reach a single state in level m 1, for all m A QSF process is a RES process if and only if one step up transitions from a level m can only reach a single set of states in the highest level M 2, largest of all m We derive explicit solutions and simple truncation bounds for the steady state probabilities of both DES and RES processes, when in addition Q insures ergodicity DES and RES processes have applications in many areas of applied probability comprising computer science, queueing theory, inventory theory, reliability and the theory of branching processes To motivate their applicability we present explicit solutions for the well-known open problem of the M/Er/n queue with batch arrivals, an inventory model, and a reliability model Key words : Successively Lumpable, Stochastic Processes, Queueing, Inventory, Steady State Analysis 1

2 2 1 Introduction Consider a Markov chain with states that can be specified by tuples of the form (m, i), where m Z represents the current level of the state and i Z + the current phase of the state The process is called quasi skip free (QSF) to the left (down) when its transition rate matrix Q (cf Eq (1)) does not permit one step transitions to states that are two or more levels away from the current state in the downwards direction of the level variable m QSF processes generalize the well studied class of quasi birth and death processes (QBD) for which one step transitions to states that are two or more levels away from the current state in either direction of the level variable m are not allowed QSF to the right (up) process can be defined with an apparent modification of the definition of the transition rate matrix Q In the sequel for simplicity we will refer to a QSF process when the skip free direction is apparent (and without loss of generality taken to be the down direction) This article defines and studies the down entrance state (DES) and the restart entrance state (RES) classes of QSF processes specified in terms of the nonzero structure of the elements of the matrix Q Explicitly: a) a QSF process satisfies the (DES) property if and only if one step down transitions from a level m can only reach a single state in level m 1, for all m and b) a QSF process satisfies the (RES) property if and only if one step up transitions from a level m can only reach a single set of states in the highest level M 2, largest of all m The main contributions of this paper are as follows First, we derive explicit solutions for the steady state probabilities of level dependent DES and RES processes, cf Eqs (10), (13), (26) and (29) Second, we use state truncation to derive tight bounds for the steady state probabilities Third, we show that the DES property is satisfied by many models that arise in practice and we obtain explicit solutions for the well-known open problem of the M/Er/n queue with batch arrivals We note that to our knowledge there are no explicit expressions in the literature for the rate matrix set (and hence for the invariant measure) of this model This is due to the QSF structure We also demonstrate the applicability of DES processes to an inventory model with random yield Finally we discuss the restart hypercube model as an example of a RES process

3 3 Although our methodology applies to infinite values for the number of levels in the process in the downward and/or in the upward direction as well as for the number of phases, we take up explicitly only the case of finite values so that we can employ finite matrices in the analysis In this way the basic features of the theory will not be obscured by additional formalism However, we want to emphasize that all the results generalize to infinite values in a natural way using truncation, cf Section 32 herein and Vere-Jones (1967), Tweedie (1973) and Seneta (1980) We will allow the lowest level value to be negative in order to have a natural state description for eg inventory models where shortage is allowed We also note that the results regarding the explicit expressions for the rate matrix set hold for the case that one boundary-side (up or down side) of the state space is a transient class of states However, for simplicity we will not consider this case The smaller class of level homogenous QBD (LHQBD) processes cf Neuts (1981), has been used to model systems in many areas including queueing theory, cf Riska and Smirni (2002), retrial queues, cf Artalejo et al (2010) For algorithmic usages of QBDs we refer to the anti-plagiarism scanner software of Viper We note that most of the early literature devoted to level homogeneous QBDs typically follows the approach presented in Chapter 6, p 129, in Latouche and Ramaswami (1999), whereby the computation of the steady state distribution is based on computing a rate matrix R This matrix is specified as one of the solutions of the, not easy to solve, matrix quadratic equation: R 2 D + RW + U = 0, where we use our current (cf Eq (1)) notation: D, W, U for the matrices A 2 A 1, A 0 in Latouche and Ramaswami (1999) Numerical methods for computing R involve cyclic reduction Bini and Meini (1996) and logarithmic reduction Pérez and van Houdt (2011), Latouche and Ramaswami (1993) For instance R is expressed in terms of a matrix G (such that R = U(I W UG) 1 ) which is the solution of matrix quadratic equation: UG 2 +W G+D = 0 For recent work on numerical methods for computing the matrix G for QBD processes of special structure we refer to van Houdt and van Leeuwaarden (2011), Etessami et al (2010), Bini et al (2005) and references therein A general approach to compute G, exists only in special cases when the down transition matrix has the form c r with c a column vector and r a row vector normalized to one We note that the DES property is implied by the c r structure but in our present setting we deal with the much more general class of QSF processes

4 4 In Latouche and Ramaswami (1999), Chapter 13, p , a method is given to analyze level homogeneous QSF (LHQSF) processes by considering them as embedded processes in suitably defined QBDs However, as stated therein, the success of this approach has been limited We note that the matrix analytic method has been used in Ramaswami (1988) to derive a recursive solution for the M/G/1 queue which is a QSF process, using matrix G described above In Bright and Taylor (1995) recursive algorithms are given to compute the rate matrices for level dependent QBD processes (LDQBD) We note that when this model satisfies the DES property then our Eq (8) and Eq (9) provide a rate matrix set explicitly The classes of DES and RES processes have the capacity to model many complex problems, such as Lin et al (2009), Guillemin et al (2004), Zwart and Boxma (2000), Gaver et al (2006) and Baer et al (2013) Indeed in this paper we obtain explicit solutions for a well-known queueing problem In addition, RES processes have been used to represent restart systems, cf Katehakis and Veinott Jr (1987), Tong et al (2006) and Sonin (2011), chains that represent inventory systems with random yield or lead times, reliability, cf Kapodistria (2011), and computer science and the theory of branching processes, cf Brown et al (2010) The rest of the article is organized as follows In Section 2 we formally define a QSF process and show that the DES property implies a successive lumpable property for a QSF process, cf Katehakis and Smit (2012) Then, in Theorem 2 we provide explicit expressions for the a rate matrix set, that provides a recursive relation for the steady state probabilities between a level of the state space and its sub-levels These expressions readily yield the state state probabilities as shown in Theorem 3 In Section 32 we show how the state space can be truncated, both in the downwards as in the upward directions In Section 5 we show how this methodology can be applied to the M/Er/n queue with batch arrivals, and to an inventory model with random yield We note that the analysis of these models readily extends to the case of the Phase type distributions, we consider the Er distributions for simplicity of the exposition In the Appendix we give a proof of Theorem 1

5 5 2 Basic Notation for DES and RES Processes A QSF to the left (or down ) process is a continuous time Markov process X(t) on state space X that can be expressed as X = M 2 m= {(m, 1), (m, 2),, (m, l m )}, where l m,, M 2, are some fixed finite integers, with 1 l m <, < < M 2 <, and m M 2 A state (m, i) specifies its current level m and its within the level state i, with i = 1,, l m For a QSF process, the transition rate matrix has the form: W U, +1 U,m U,m+1 U,M 2 1 U,M 2 D 1 W +1 U +1,m U +1,m+1 U +1,M 2 1 U +1,M W Q = m U m,m+1 U m 1,M 2 1 U m 1,M D m+1 W m+1 U m,m 2 1 U m,m, (1) W M 2 1 U M 2 1,M D M 2 W M 2 where in the above specification of Q we use the notation D m, W m, and U m,k describe respectively the down (to level m 1), within (level m), and up (to level k = m + 1, m + 2,, M 2 ) transition rate sub-matrices in relation to the current level m of a state (m, i) The dimensions of these matrices are respectively l m l m 1, l m l m and l m l k ; the 0 sub-matrices of Q have position dependent dimensions so that Q is a well defined transition rate matrix Note that in the special case that there exist matrices D, W, U k such that D m = D, W m = W and U m,k = U k, for all m, the process is called (level) homogeneous When U m,m+k are all 0 matrices for all k 2 and all m, the QSF process reduces to the well studied QBD process, cf Artalejo and Gómez-Corral (2008) In the sequel we will assume that the QSF processes discussed are ergodic Explicit sufficient conditions for the elements of Q can be derived to support this claim, cf Remark 3; such conditions on ergodicity for multi dimensional Markov Chains can be found in Szpankowski (1988) and various criteria in Tweedie (1981), in Hordijk and Spieksma (1992) and in Spieksma and Tweedie (1994) to

6 6 We consider Markov process X(t), introduced above Clearly the state space X of this process can be partitioned into a (possibly infinite) sequence of mutually exclusive and exhaustive level sets: L m = {(m, 1), (m, 2),, (m, l m )} Definition 1 For any fixed m we define the sub-level set of L m to be the set of states L m = m k= L k while the set L m = M 2 k=m L k is the super-level set of L m We let π(m, i) denote the steady state probability of state (m, i) The vectors π m := [π(m, 1),, π(m, l m )] and π m := [π,, π m ], will denote respectively the steady state probabilities of states in level L m and sub-level L m The vector of the steady state probabilities over all states will be denoted by π := [π,, π M 2] = π M 2 Using the QSF structure of Eq (1) of the rate matrix Q, the (potentially non-zero) elements of the matrices D m, W m, U m,k will be denoted respectively by d(m 1, j m, i), (a down rate), w(m, j m, i) (a within rate) and u(k, j m, i), (an up rate) for k > m Note that the diagonal elements of W m are the negative sum of all other elements in the m row of Q The DES and RES processes we consider in the sequel are are level dependent and ergodic QSF processes with a transition rate matrix Q that satisfies one of the two properties stated in Definition 2 These conditions follow from the structure of the nonzero elements of Q Definition 2 i) The down entrance state (DES) property: one step down transitions from a level m can only reach a single state in level m 1, for all levels m ii) The restart entrance state (RES) property: one step up transitions from any level m can only reach a single state in level M 2 (M 2 is the highest level) for all levels m 3 The Class of DES processes The following lemmas show that the simple algebraic characterization of the DES property is a sufficient condition for a QSF process to be successively lumpable Following Katehakis and Smit (2012), a state is an entrance state of a subset X 0 of the state

7 7 space X if all one step transitions from outside this set X 0 into X 0 can only occur via a transition to the entrance state We start with the following Lemma which characterizes an entrance state of a sublevel set L m of a QSF process in terms of an algebraic property of the down transition sub-matrix D m of its transition rate matrix Q Lemma 1 For a QSF process X(t) and for a fixed m {,, M 2 }, a state (m, ε(l m )) L m is an entrance state for L m if and only if the following is true for all (m + 1, i) L m+1 : d(m, j m + 1, i) = 0, if (m, j) (m, ε(l m )) (2) Proof The structure of the rate matrix Q implies that down transitions leaving the set L m+1 = L m+1 L m+2 can only come from states in L m+1 Further, by Eq (2) the latter type of transitions are possible only when they lead into the same state (m, ε(l m )) L m It is easy to see that Eq (2) of Lemma 1 is equivalent to the statement that the l m l m 1 matrix D m has a single nonzero column For any fixed n {,, M 2 }, let D n denote the partition {L n, L n+1,, L M2 } of X For a fixed n, the next lemma establishes that when D m has a single non-zero column for all m n + 1, ie, Q has the DES property, then the QSF process is successively lumpable with respect to the partition D n Lemma 2 A QSF process is successively lumpable with respect to a partition D M1 D m contains a single non-zero column vector for all m = + 1,, M 2 if Proof It is a direct consequence of Lemma 1, that for a QSF process, a sub-level set L m has an entrance state (m, ε(l m )) if D m+1 contains a single non-zero column vector This is true for all m Since L m = L m 1 L m it follows from the definition that the chain is successively lumpable: in the notation of Katehakis and Smit (2012), D 0 corresponds to L and for m > : D m corresponds to L m M1 Note that when a QSF process is successively lumpable with respect to a partition D n then it is successively lumpable with respect to partition D m for all m > A

8 8 L m 1 L m L m+1 m 1, 1 m, 1 m + 1, 1 m 1, lm 1 m, lm m + 1, lm+1 Figure 1 Graphical representation of a Successively Lumpable QSF process graphical representation of the transitions that are allowed in a successively lumpable QSF process can be found in Figure 3 We next state the following assumption that will be used in the sequel, where for notational simplicity we let the entrance state of a set L m be state (m, 1), for all m without loss of generality Assumption 1 The DES process has a transition rate matrix Q with the following properties: A1 For all m { + 1,, M 2 }, only the first column of sub-matrix D m contains non-zero elements, ie, d(m 1, 1 m, i) > 0 for at least one (m, i) L m and all other columns of D m are equal to zero A2 The QSF process is ergodic (irreducible) A3 The QSF process has bounded rates Remark 1 Part (A3) of Assumption 1 is given to make the proofs applicable for an infinite state space and is not strictly necessary It can be relaxed without further conditions; it has been added to the assumption to make some of the proofs easier to state

9 9 31 Explicit Solutions We will express the steady state distribution of states in L m in that of L m The presence of entrance states guarantees that sojourn times in L m are independent of the rates from states in L m We will use the the following notation We first define the scalar l m := m k= l k and use the symbols I m and Ĩm for the identity matrices of dimension l m l m and lm l m, respectively Second we define the (row)vectors of dimension l m 0 m, 1 m and δ m to represent a vector identically equal to 0, a vector identically equal to 1, a vector with 1 as its first coordinate and 0 elsewhere respectively Next we define the (row)vectors of dimension l m : 0 m and 1 m to be the vectors will all its coordinates equal to 0 and 1, respectively Finally, we define the matrix Ũ m,n of dimension l m l m by: M 2 Ũ m,n = U m,k 1 kδ m, (3) k=n where the elements of Ũ m,n will be denoted by ũ(m, i), thus, M 2 ũ(m, i) := l k k=n j=1 u(k, j m, j) We next define the rate sub-matrices: Ũ,m+1 + U,m A m =, (4) Ũ m 1,m+1 + U m 1,m B m = Ũ m,m+1 + W m (5) and Note that: [ ] Am Γ m := B m i) The matrix A m contains all rates of Q corresponding to transitions from states in L m 1, into states of L m plus rates corresponding to transitions under Q from states in L m 1 into states of L m+1, which under A m have been re-directed to transitions into the entrance state (m, 1)

10 10 ii) The l m l m matrix B m contains all rates of Q corresponding to transitions from states in L m, into states of L m plus rates corresponding to transitions under Q from states in L m into states of L m+1, which under B m have been re-directed to transitions into the entrance state (m, 1) Thus, for m > since the construction of B m excludes all down transitions it a transient transition rate matrix However, by the definition of Ũ, +1 and by its construction the matrix B M1 is an l M1 l M1 conservative transition rate matrix We next state and prove a proposition regarding basic properties of B m Proposition 1 The following are true: i) The matrix B M1 is irreducible ii) The matrices B m are non-singular, for all m > iii) The inverse of B m has non-positive elements, for all m > Proof To prove i), we will show that every state (, j) in level L M1 that is not the entrance state (, 1) communicates with state (, 1) First, recall that in the construction of B M1, up -transitions are redirected to the entrance state If there are no up -transitions from the communicating class containing (, j), this class would be a closed class in Q which would not contain state (, 1), a contradiction to the irreducibility assumption of Q So the entrance state is reachable from (, j) under B M1 Second, we show that (, j) is reachable from state (, 1) under B M1 Indeed, the only way to reach (, j) from a state in L M1 +1 is via the entrance state, and such a path has to exist by irreducibility of Q Since the within L M1 transition rates under Q are all preserved under B M1, state (, j) is reachable from state (, 1) under B M1 Thus every state communicates with the entrance state, and we conclude that B M1 is irreducible For ii), we will call a matrix diagonally dominant if the absolute value of a diagonal elements are greater or equal than the sum of the absolute values of the off diagonal elements in that row, and strict inequality holds for at least one row An irreducible diagonally dominant matrix is non singular by the well-known Levy-Desplanques theorem, cf Varga (1963) (p 85) or Varga (1976) The construction of B m excludes all down transitions and that makes it a diagonally dominant matrix So when B m is irreducible, the claim of the lemma is true

11 However, in general it is possible that the matrix B m is not irreducible (even though the matrix Q is irreducible) In this case we will show that the construction of B m with the irreducibility of Q implies the non-singular property of B m Indeed, suppose B m contains two or more communicating classes of states, say C 1,, C km, where C e contains the entrance state (m, 1) We will relabel the states such that states in the same class have adjacent indices and such that if a state in a class C i has a transition to a state in a class C j then i < j It is clear that this relabeling is feasible and we can write B m as: Z 11 Z 12 Z 13 0 Z 22 Z 23 B m = 0 0 Z 33, where Z ii is a matrix of size C i C i containing transition rates within C i and the matrix Z ij is of size C i C j containing transition rates from C i to C j (and is possible to have some nonzero elements) We next show that the determinants of Z ii are all non zero This is sufficient to show that B m is non-singular, since its determinant is the product of the determinants of the Z ii s First, at least one state in C e has to have a transition down or to another class C j within B m, since otherwise C e would be part of a closed absorbing class under Q This class does not contain any states in L m and this is a contradiction to the irreducibility of Q Thus Z ee is diagonally dominant and therefore non-singular Furthermore, any other class C j (j e) has to have a state that has a transition rate leaving this class (ie, down, up or to another class C j (j j)) otherwise C j would be a closed class under Q Therefore, the sum of the off diagonal elements of Z jj in at least one of its row is strictly less than the absolute value of the corresponding diagonal element, since at least one transition leads to a state out of C j Thus Z jj is diagonally dominant for all j and the proof of part ii) is complete For iii), let τ i denote the maximum of the absolute values of the diagonal elements of Z ii and let Γ i = τ i I + Z ii Since τ i is finite and positive and Γ i is non-negative with row sum less or equal to τ i, (and strictly less than τ i for at least one row), the row sum of τ 1 i Γ i is smaller or 11

12 12 equal to one for each row This implies that τ 1 i Γ i is a transient transition matrix with spectral radius smaller than 1, and thus all elements of j=0 1 (τi and finite, see for example Seneta (1981), Theorem 43 Note also that (Z ii ) 1 = (τ i (τ 1 i Γ i I)) 1 = τ 1 i (τ 1 i Γ i I) 1 = τ 1 The above implies that all elements of (Z ii ) 1 are non-positive By Woodbury s identity, cf Woodbury (1950) we know: [ ] 1 Z11 Z 12 = 0 Z 22 [ Z 1 11 Z 1 11 Z 12 Z Z 1 22 i ], j=0 Γ i ) j are non-negative (τ 1 i Γ i ) j which is a non-positive matrix by the above; an induction argument can establish the same for (B m ) 1 We next state the following theorem for successively lumpable Markov chains within the context and notation of the DES model This theorem is a consequence of Theorem 2 of Katehakis and Smit (2012); to avoid the introduction of the notation used in the latter paper, we provide a separate proof in the appendix Theorem 1 Under Assumption 1, the following equality is true for the steady state probabilities π m of X(t) for every m : π m Γ m = 0 m (6) We next introduce the idea of a rate matrix set for Q as a sequence of matrices R = {R k m} m,k such that R k m satisfy Eq (7), for all k = 1,, m and m = +1,, M 2 ; cf Latouche and Ramaswami (1999) and references therein π m = π m k R k m (7) Note that there are multiple rate matrix sets, for a given Q To see this note that for any fixed k, m and known vectors π m = [π(m, 1),, π(m, l m )] and π m k = [π,, π m k ] Eq (7) is essentially a system of l m equations with l m k l m unknowns, the elements of the matrix R k m These equations have many solutions In Theorem 2 we show that the specific set R 0 := {R k m} m,k obtained recursively using Eq (8) starting with Eq (9), is a rate matrix set for Q For all m = + 1, M 2 with k = 2, m we define:

13 13 R k m := [ ] Ĩ m k R 1 m (k 1) R k 1 m, (8) where: R 1 m := A m (B m ) 1 (9) By virtue of Proposition 1ii we know that B m is non sinular Theorem 2 The set R 0 defined by Eqs (8) and (9) above is a rate matrix set for Q Proof The proof is by induction For k = 1 we know by Theorem 1 that π m Γ m = 0 m We can rewrite this as: and thus: [ ] [π m 1 π m Am ] = 0 B m, m π m = π m 1 A m (B m ) 1 = π m 1 R 1 m Suppose the statement is true for any m and for k 1 We next show that the statement holds for k: π m = π m (k 1) R k 1 m = [π m k π m (k 1) ]R k 1 m = [π m k π m k Rm (k 1)]R 1 k 1 m = [ ] π m k Ĩ m k R 1 m (k 1) R k 1 m = π m k R k m Thus the statement is true for k = 1,, m and therefore: π m = π m k R k m Note that the above implies that we can express all vectors π m in terms of the steady state distribution of level, since is finite, By the irreducibility assumption all vectors are strictly larger than 0 Therefore we state: π m = π R m m > 0 m, (10)

14 14 For any m 1, m 2 {,, M 2 }, with m 1 < m 2, we define the column vector S m 2 m 1 length l m1 Remark 2 by Eq (11) below S m 2 m 1 = [ 1 m 1 + m 2 m=m 1 +1 R m m 1 m 1 m ] of (11) The elements of R k m are non-negative k, m where + 1 m M 2, 1 k m To check this claim for R 1 m, it suffices to note that A m 0 by definition and B 1 m > 0, by Proposition 1 Alternatively, the (i, j)-th element of R 1 m can be given an expected first passage time interpretation as is described for QBD processes in Latouche and Ramaswami (1999), chapter 6 The claim for R k m, with k 2, follows using Eq (8) The lemma below establishes the relation between π and S M 2 Lemma 3 The following relation holds for π and S M 2 : π S M 2 = 1 (12) Proof Since the chain is ergodic we have π 1 + M 2 m= +1 πm 1 m = 1, thus Eq (10) implies: Substituting [ 1 + M 2 π [ 1 + m= +1 Rm m 1 m M 2 m= +1 We now state and prove the following theorem ] R m m 1 m Theorem 3 Under Assumption 1, the following is true: ] = 1 by S M 2 in the above gives Eq (12) π = δ M1 [ S M 2 δ M1 B M1 ] 1 (13) Proof Since B M1 is an irreducible rate matrix (Proposition 1 i), it has rank (l M1 1) by basic linear algebra theory, see for example Seneta (1981) Furthermore, at the boundary we have Γ M1 = B M1 and π = π Thus Theorem 1 implies that Eq (6) can be written as Eq (14) below π B M1 = 0 M1 (14) Thus it follows that the vector S M 2 is not an element of the linear space spanned by the columns of B M1, since by Lemma 3 π S M 2 = 1

15 The above imply that B M1 has rank (l M1 1) and when we add to its first column the vector S M 2 the resulting matrix [S M 2 δ M1 B M1 ] has full rank and it is invertible We use Lemma 3 to state [ π S M 2 ] δm1 = δ M1, 15 and via π [ S M 2 δ M1 B M1 ] = δm1 0 M1 = δ M1, we conclude π = δ M1 [ S M 2 δ M1 B M1 ] 1 The results above justify the following algorithm to find the steady state distribution of a DES-QSF process Algorithm 1 (DES-QSF) - Calculate R 1 m with Eq (9) for all m = + 1,, M 2 - Compute R k m recursively via Eq (8) for m = + 1, M 2 and k = 2,, m - Calculate S M 2 via Eq (11) - Calculate π via Eq (13) - Calculate π m via Eq (10) for all m = + 1,, M 2 32 State Space Truncations In this section we show how to truncate the state space in the upward direction, in order to obtain upper bounds for the steady state probabilities π(m, i) of states in L m 2 where m 2 {, + 1,, M 2 1} To this end we first define a process X m2 (t) with truncated state space X m2 = L m 2 and transition rate matrix: Q Xm2 = D m 2 2 W m 2 2 U m 2 2,m 2 1 U m 2 2,m 2 + Ũ m 2 2,m 2 0 D m 2 1 W m 2 1 U m 2 1,m 2 + Ũ m 2 1,m 2, (15) 0 0 D m 2 W m 2 + Ũ m 2,m 2 where the elements of the last column are given by Eq (3) We denote the steady state distribution of this process as the row vector π Xm2 = [π Xm 2,, π m 2 Xm 2 ] of size:

16 16 m2 m= l m, where its m th component contains the steady state probabilities for level m of the truncated process We next state the following We emphasize that this proposition clearly holds for M 2 = under the ergodicity assumption Proposition 2 For all finite m 2, and any level m =, + 1,, m 2, the following are true: i) π m Xm 2 = π Xm 2 R m m, (16) ii) π Xm 2 = δ M1 [ S m 2 δ M1 B M1 ] 1 (17) π(m, i) < π Xm2 (m, i) iii) For all states (m, i), π Xν (m, i) is a strict decreasing function in ν = m 2, m 2 + 1, Proof By its construction, the process X m2 (t) is a QSF process which satisfies Assumption A1 Further, by its definition the matrix Q Xm2 gives rise to the same rate matrices R k m as the matrix Q of the original process X(t) This follows from the fact that this specific truncation ensures that the matrices A m,xm2, B m,xm2 of the truncated process corresponding to the matrices A m B m of the original process are identical and this proves Eq (16) The proof of Eq (17) follows as the proof of Theorem 3, if we replace M 2 with m 2 For the proof of part ii), using Proposition 1 of Katehakis and Smit (2012), (where π Xm2 (m, i) = v L m(m, i)) we obtain that Eq (18) below is valid for all m ν: π Xν (m, i) = π(m, i) (k,j) L ν π(k, j) for all ν = m 2, m 2 + 1, (18) Since π(k, j) < 1, for all finite ν, it follows from the above that π(m, i) < (k,j) L ν π Xm2 (m, i) For the proof of part iii), note that since L ν L ν+1, we have that π(k, j) < (k,j) L ν π(k, j) Thus, we conclude that π (m, i) < π (k,j) L ν+1 Xν+1 Xν (m, i) We can repeat this argument for L ν+2, L ν+3, and the proof is complete Note that Proposition 2 is closely related the results of Bright and Taylor (1995) (pp ), derived for LDQBDs Specifically, Eq (16)-(17) are the QSF process extensions of Eqs (17)-(18) of that paper, with k and m reversed and the change of notation K, x k, R k+1, R 0 in place of our m 2, π m, R m R 1

17 Note that the matrix R m m is finite even when the the QSF process is not ergodic; such a non-ergodic case exists for instance when there is a drift to up direction We can however state the following: Remark 3 The successively lumpable QSF process is ergodic if M 2 m= R m m <, since then Theorem 3 and Eq (10) show that there exist positive steady state probabilities for all states Similarly, it follows that for QSF processes to be ergodic it is sufficient that S M 2 < Remark 4 One can also construct truncations with respect to or to any l m separately This is especially important when some of these constants are infinite There are various truncations methods possible to truncate the matrix Q of infinite size, some are described in Vere-Jones (1967), Seneta (1980) and Tweedie (1973) Most of these truncations will preserve the successively lumpable property Such as truncation to m 1 = we provide below, following Seneta (1980) Define a process X m1 (t) with state space X m1 = L m1 and transition rate matrix: W m 1 U m 1,m 1 +1 U m 1,m 1 +2 U m 1,m 1 +3 D m 1+1 W m 1+1 U m 1+1,m 1 +2 U m 1+1,m+3 Q Xm1 = 0 D m 1+2 W m 1+2 U m 1+2,m 1 +3, (19) where w(m 1, j m 1, i) = w(m 1, j m 1, i) if j 1 and otherwise we have w(m 1, 1 m 1, i) = w(m 1, 1 m 1, i) + d(m 1 1, 1 m 1, i) Let π Xm1 17 denote the steady state distribution of X m1 (t) For this truncation it is shown in Seneta (1980) (Theorem on p 262) that for all (m, i): 4 The Class of RES processes π(m, i) = lim π Xm m 1 1 (m, i) In this section we consider QSF processes of the form described in Definition 2ii Because the current section has the same structure as Section 31, most of the theorems and lemmas can be given without proof We will refer to the corresponding statement in the previous section and add some clarification when necessary We start with the following lemmas that show that the simple algebraic characterization of the RES property is a sufficient condition for a QSF process to be successively lumpable

18 18 We start with the following lemma that characterizes an entrance state of the superlevel set L m of a QSF process in terms of an algebraic property of the up transition sub-matrices U m,k of its transition rate matrix Q Lemma 4 For a QSF process X(t) and for all levels m {,, M 2 }, the state (M 2, ε(l m )) L M2 is an entrance state for the set L m if the following is true for all states (n, i) L m 1: u(k, j n, i) = 0, if (k, j) (M 2, ε(l m )) (20) Proof Similar to statement made in the proof of Lemma 1 that regards DES processes It is easy to see that Eq (20) of Lemma 4 is equivalent to the statement that the matrices U m,m 2 have a single nonzero column and that U m,k = 0 for all k {m + 1,, M 2 1} This is equivalent to the RES property For any fixed n {,, M 2 }, let D n denote the partition {L M1,, L n 1, L n } of X For a fixed n, the next lemma establishes that when Q has the RES property (cf 2 ii), then the QSF process is successively lumpable with respect to the partition D n Lemma 5 A QSF process is successively lumpable with respect to a partition D M2 if the matrices U m,m 2 have a single nonzero column and that U m,k = 0 for all k {m + 1,, M 2 1} Proof Similar to the proof of Lemma 2 in the previous section A graphical representation of the transitions that are allowed in a RES QSF process can be found in Figure 2 We now state the following assumption that will be used in the sequel of this section, where for notational simplicity we let the entrance state of a set L m be state (M 2, 1), for all m without loss of generality Assumption 2 The RES process under consideration has a transition rate matrix Q with the following properties: A1 The process is ergodic (irreducible);

19 19 L m 1 L m L M2 m 1, 1 m, 1 M2, 1 m 1, lm 1 m, lm M2, lm 2 Figure 2 Graphical representation of a RES QSF process A2 For all m {,, M 2 1}, only the first column of sub-matrix U m,m 2 can contain non-zero elements A3 The RES process has bounded rates Since the process is positive recurrent, there has to be a transition back to the entrance state from a state in L m, ie u(m 2, 1 n, i) > 0 for at least one (n, i) L m and all other columns of U m,m 2 {m,, M 2 1} are equal to zero In addition, U m,k = 0 for all k 41 Explicit Solutions We will use the notation introduced in the previous section In addition we define the matrix D m of dimension l m l m by: D m = D m 1 mδ m (21) We next state and prove a proposition regarding basic properties of W m and D M 2 Proposition 3 The following are true: i) The matrix W M 2 + D M 2 is irreducible ii) The matrices W m are non-singular, for all m {,, M 2 } iii) The elements of the inverse of W m are non positive, for all m < M 2 Proof The proves are analogous to that of Proposition 1 in the previous section, that states similar results for a DES process

20 20 We can now state the following theorem for RES process, within the context and notation of the present section Theorem 4 Under Assumption 2, the following equality is true for the steady state probabilities π m of X(t) for every m {,, M 2 1} : π m W m + π m+1 D m+1 = 0 m (22) π M 2 (W M 2 + D M 2 ) = 0 M2 (23) Proof The proof is along the same lines as Theorem 1, since the RES process is successively lumpableas well We can complete the proof by only considering the states that have possibly positive transitions out of set L m+1 For a RES process, we define a rate matrix set for Q as a sequence of matrices R = {R m } m such that R m satisfy Eq (24), for all m =,, M 2 1 Note that this is a slightly different definition than the one introduced for DES processes π m = π m+1 R m (24) In Theorem 5 we show that the specific set R 0 := {R m } m given by Eq (25) below, is a rate matrix set for Q For all m =,, M 2 1 we define: R m := D m+1 (W m ) 1 (25) By virtue of Proposition 3ii we know that W m is non singular Theorem 5 The set R 0 defined by Eq (25) above is a rate matrix set for Q Proof Follows directly from Theorem 4 Note that the above implies that we can express all vectors π m in terms of the steady state distribution of elements in level M 2, since M 2 is finite By the irreducibility assumption all vectors are strictly larger than 0 Therefore we state: π m = π M 2 M 2 1 m k=0 R M2 1 k > 0 m, (26)

21 l M2 For any m 1 {,, M 2 }, with m 1 < M 2, we define the column vector T M 2 m 1 by Eq (27) below T M 2 m 1 = [ 1 M 2 + M 2 1 m=m 1 Note that R m is non-negative for all m M 2 1 m k=0 R M2 1 k1 m The lemma below establishes the relation between π M 2 and T M 2 Lemma 6 The following relation holds for π M 2 and T M 2 : ] 21 of length (27) π M 2 T M 2 = 1 (28) Proof Analogous to Lemma 6 We now state and prove the following theorem Theorem 6 Under Assumption 2, the following is true: π M 2 = δ M2 [T M 2 δ M2 W M 2 D M 2] 1 (29) Proof Since W M 2 D M 2 is an irreducible rate matrix, Proposition 3 i, it has rank (l M2 1) by basic linear algebra theory, see for example Seneta (1981) Furthermore, we know that π M 2(W M 2 D M 2) = 0 M2 and π M 2T M 2 = 1, thus that the vector T M 2 not an element of the linear space spanned by the columns of W M 2 D M 2 Therefore [T M 2 δ M1 W M 2 D M 2] has full rank and is invertible The remainder of the proof is similar to the proof of Theorem 3 The results above justify the following algorithm to find the steady state distribution of a RES QSF process is Algorithm 2 [RES-QSF] - Compute R m via Eq (25) for m =, M Calculate T M 2 via Eq (27) - Calculate π M 2 via Eq (29) - Calculate π m via Eq (26) for all m =,, M 2 1

22 22 42 State Space Truncations In this section we show how to truncate the state space of a RES QSF process in the downward direction in order to obtain upper bounds for the steady state probabilities π(m, i) of states in L m1 where m 1 { + 1, + 2,, M 2 } To this end we define a process X m1 (t) with truncated state space X m1 = L m1 and transition rate matrix: W m Dm 1 + U m 1,M 2 D m 1+1 W m U m 1+1,M 2 Q Xm1 = (30) 0 0 D M 2 2 W M U M 2 2,M D M 2 1 W M 2 1 U M 2 1,M D M 2 W M 2 We denote the steady state distribution of this process as the row vector π Xm1 = [π m 1 Xm 1,, π M 2 Xm 1 ] of size: M 2 m=m 1 l m, where its m th component contains the steady state probabilities for level m of the truncated process We next state the following We emphasize that this proposition clearly holds for = under the ergodicity assumption Proposition 4 For all finite m 1 M 2, and any level m = m 1, m 1 + 1,, M 2, the following are true: i) π m Xm 1 = π M 2 Xm 1 ii) M 2 1 m k=0 R M2 1 k (31) π M 2 Xm 1 = δ M2 [ T M 2 m 1 δ M2 W M 2 D M 2] 1 (32) π(m, i) < π Xm1 (m, i) iii) For all states (m, i), π Xν (m, i) is a strict decreasing function in ν = m 1, m 1 1, Proof Similar to the proof of Proposition 2: the RES property remains intact, the rate matrices do not change The entrance state will never be removed from the state space Remark 5 It is not useful to truncate the process in the upward direction Since we consider a ergodic process with a restart entrance state, returns will go to the highest level Removing this set would effect the structure of the process too much to give any bounds of interest

23 23 5 Applications To illustrate the application of the results we provide explicit solutions and approximations to well known open problems of queueing, cf Adan et al (2013), and to a stochastic inventory theory problem, cf Veinott (1965) The queueing models under consideration are the M/Er/n queueing model with batch arrivals and the Er/M/n queueing model In the two subsequent sections, we take the number of phases to be constant, ie l m = l for all m, this is done solely for presentation simplicity The analysis is easy to extend when the number of phases of the corresponding distribution is a function of m - the state of the queue, the number of customers in line The steady state distribution of the M /Er/n is known for the case of Poisson arrivals, as for example discussed in Latouche and Ramaswami (1999) As far as we known this is the first time the steady state distribution of the M/Er/n model with batch arrivals is obtained We note that the same book gives an exact solution procedure for QBD processes only when is finite Below we show that our direct method works for the Er/M/n queueing system, ie, we provide explicit formulas for the rate matrix set, even when = For the inventory model we show that is has the same structure as the M /Er/n and it can be handled similarly The construction in the following remark can be used to extend the applicability of the methods described in the previous section to models that are QSF and successively lumpable in the upward direction Remark 6 Consider a process with a transition rate matrix Q that has the block form shown in Eq (33), where its elements are labeled by (m, i) X, the states of the underlying process D m 1,m 2 W m 1 U m Q = D m,m 2 D m,m 1 W m U m 0 (33) D m+1,m 2 D m+1,m 1 D m+1,m W m+1 U m+1 Then, we can construct a transition rate matrix ˆQ of the form of Eq (1) by relabeling the states so that a new state ( m, i) corresponds to the original (m, i) X by redefining the down, within and up sub-matrices of ˆQ as follows: ˆD m = U m, Û m, k = D m,k, and Ŵ m = W m The steady state probabilities of the Q-process can be readily obtained from those of the ˆQ-process

24 24 51 The Classic M/Er/n Model with Batch Arrivals In a M/Er/n queueing system with batch arrivals the service of a customer occurs in l phases, each exponentially distributed with parameter µ i for the i-th phase of the service For notational simplicity of the exposition we describe in detail the case in which a batch may contain either 1 or 2 customers Batches with a single customer arrive according to a Poisson process with rate pλ m,i when there are m customers in the system and the served customer has gone through the first i phases of services Similarly, batches of 2 customers arrive with rate (1 p)λ m,i with p [0, 1] The service of a customer has to be completed before another customer can start his first phase In order to have state notation that is consistent with that of Section 2, we use the following state description For i < l, state (m, i) denotes the event that there are m customers in the waiting line of the system and a customer in service that has gone through i phases of the service State (m, l) denotes the event that there are m customers in the waiting line and a service completion has just occurred, so that one of the waiting customers is starting service Note that with this awkward but convenient notation the empty state of the system is state (0, l) Then, it is easy to see that this system can be modeled as a QSF process Its state space is X = {L 0, L 1,, L M2 } with L m = {(m, 1),, (m, l)} and M 2 The Q matrix is defined by Eq (1), with U m, W m and D m (all of size l l) as given below and for m 1 : λ 0,1 µ 1 µ λ 0,2 µ 2 µ 2 0 W 0 =, 0 0 λ 0,l 1 µ l 1 µ l λ 0,l λ m,1 µ 1 µ λ m,2 µ 2 µ W m =, D m = λ m,l 1 µ l 1 µ l λ m,l µ l µ l For m = 0, 1, : λ λ m, m, λ m, λ m,2 0 0 U m,m+1 = p 0 0 λ m,3 0, U m,m λ m,3 0 = (1 p) λ λ m,l m,l 0,

25 (1 p)λ 1,l 25 L 0 L 1 L 2 (1 p)λ 0,l 0, l pλ 0,l 1, l pλ 1,l 2, l pλ 2,l µ l 1 µ l 1 (1 p)λ 0,2 µ l 1 (1 p)λ 1,2 0, 2 pλ 0,2 1, 2 pλ 1,2 2, 2 pλ 2,2 µ 1 µ l µ 1 µ l µ 1 µ l 0, 1 pλ 0,1 1, 1 (1 p)λ 0,1 pλ 1,1 2, 1 pλ 2,1 (1 p)λ 1,1 Figure 3 An M/Er/n queueing process with batch arrivals Note that (m, 1) is the entrance state of the set L m, because D m has a single nonzero column The matrices A m and B m, described in Section 2, are: 0 m 3,m (1 p)λ m 2, (1 p)λ m 2, (1 p)λ m 2,3 0 A m = (1 p)λ m 2,l λ m 1, (1 p)λ m 1,2 pλ m 1,2 0 0 (1 p)λ m 1,3 0 pλ m 1,3 0 (1 p)λ m 1,l 0 0 pλ m 1,l where 0 m 3,m is a matrix of size l m 3 l m with a 0 at every entry For m = 1, 2, : µ 1 µ λ m,2 λ m,2 µ 2 µ 2 0 B m = λ m,3 0 λ m,3 µ 3 0 λ m,l 0 0 λ m,l µ l and µ 1 µ λ 0,2 λ 0,2 µ 2 µ 2 0 B 0 = λ 0,3 0 λ 0,3 µ 3 0 λ 0,l 0 0 λ 0,l Now we can calculate R 1 m using Eq (9): R 1 m = A m (B m ) 1, where π m = π m 1 R 1 m

26 26 Since the first l m 3 rows of R 1 m are zero (due to multiplication of (B m ) 1 with the 0 m 3,m sub-matrix of A m ) this expression reduces to the following π m = [π m 2 π m 1 ]R 1 m, where R 1 m denotes the nonzero rows of R 1 m R k m We can construct the sequence of rate matrices R k m using Eq (8) and the notation for the sub-matrix of the nonzero rows of R k m we obtain: and π m = π 0 R m m π m = [π m k 1 π m k ]R k m When M 2 is finite (ie there is a finite buffer for the number of customers allowed in the system), then Theorem 3 readily provides the solution: π 0 = δ 0 [ S M 2 0 δ 0 B 0 ] 1, π m = π 0 R m m When M 2 is infinite, using Proposition 2, we can construct upper bounds for π(m, i) via the process X m2 (t) described therein This result is stated in the next theorem Theorem 7 The following is true for the M/Er/n model with batch arrivals: where S m 2 0 = [1 0 + m 2 m=1 R m m 1 m] and Proof second claim π 0 Xm 2 = δ 0 [S m 2 0 δ 0 B 0 ] 1 (34) π m π m Xm 2 = π 0 Xm 2 R m m Directly from Theorem 3 (for Eq (34)) and Section 2 (Proposition 2) for the 52 An Inventory Model with Random Yield In this section we consider an inventory model with random yield Specifically, we investigate a system where customers arrive with rate λ and a batch of products arrives according to an exponential distribution with rate µ Random yield is possible in this model, ie the size of the batch is nl with probability p n, where l is a fixed positive constant, cf Veinott (1965) We model this inventory model process as a QSF process X(t) on state space X = {L 0, L 1, } with L m = {(m, 1),, (m, l)} In state (m, i)

27 27 there are ml + i products in stock Figure 52 displays the transition diagram of the described model with l = 3 and where the size of the batch is 3 with probability p and 6 with probability 1 p This model is a successively lumpable QSF process, where states (m, l 1) are the entrance states of sub-sets L m In this QSF process, U m,k, W m and D m are of size l l It has the same structure as the queueing model described in Section 51 and it can be solved analogously We note that we can easily obtain explicit formulas for the steady state probabilities even in the case that both λ and µ may depend on the state For example, when there is too much (little) inventory a discount (premium) price may be used for the product and this may change the arrival rate of the customers, ie, λ = λ(m, i) Also, when there is a high level of inventory one may decide not to order This can be easily incorporated in product arrival rate, ie, µ = µ(m, i) Finally we note that just as easily one can handle the extension where the batch size nl is replaced by nl m to represent dependency on the inventory level m We omitted all these dependencies in this exposition only to simplify the notation L 0 L 1 L 2 (1 p)µ (1 p)µ 0, 0 pµ 1, 0 pµ 2, 0 pµ λ λ λ (1 p)µ (1 p)µ 0, 1 pµ 1, 1 pµ 2, 1 pµ λ λ λ λ λ λ 0, 2 pµ 1, 2 (1 p)µ pµ 2, 2 pµ (1 p)µ Figure 4 An Inventory Model with random yield 53 Restart Hypercube Models In this section we illustrate the methods of Section 41 using a simplified version of this classical model for the special case of a three dimensional hypercube The extension to

28 28 the arbitrary dimension case will become apparent This type of systems have many applications in diverse fields and have been studied by many authors, cf the papers we next list and the references therein Larson (1974), Hooghiemstra and Koole (2000), Katehakis and Melolidakis (1995), Righter (1996), Koole and Spieksma (2001) and Ungureanu et al (2006) A basic version of the model is as follows, cf Derman et al (1980), Weber (1982), Nash and Weber (1982), Katehakis and Derman (1984), Katehakis (1985), Frostig (1999) A system of known structure is composed of M components and it operates continuously The time to failure of component i = 1,, M is exponentially distributed with rate µ i and it is independent of the state of the other components In this section we make the modelling assumption that when the system fails it is restored (or replaced) to a state as good as new and the time it takes for this restoration is exponentially distributed with rate λ It follows under these assumptions that at any point in time the state of the system can be identified by a boolean M- vector x = (x 1,, x M ), with x i = 1 if the i th component is working, else x i = 0 Hence X = {0, 1} M is the set of all possible states Under these conditions the time evolution of the state of the system can be described by a continuous time Markov chain The structure of the system is specified by a binary function φ defined on X Let G = {x : φ(x) = 1} denote the set of all operational (good) states of the system and let B = {x : φ(x) = 0} denote all failed states of the system For such a system it is important to compute measures of performance such as the availability of the system defined as α φ = x G π(x) Regardless of the choice of the structure φ it is easy to see that the corresponding chain is successively lumpable For example, for the parallel system we have B = {(0,, 0)} Figure 5 illustrates the transition diagram of the corresponding Markov process for the parallel system when M = 3 It is clear that this process is a RES QSF process with respect to partition D = {L 0, L 1, L 2, L 3 } of size M + 1 and x X : x L m iff x i = m It is apparent that in this RES process, = 0 and M 2 = 3 The states are ordered as is shown in Figure 5, eg, (0, 0, 1) is the first state of level L 1, (0, 1, 0) is the second state of level L 1,, (1, 1, 0) is the third state of level L 2, etc We derive that D m, W m i

29 µ1 29 λ 0, 0, 1 µ 2 0, 1, 1 µ 3 µ 3 µ1 0, 0, 0 µ 2 0, 1, 0 1, 0, 1 µ 2 1, 1, 1 µ 3 µ1 µ 3 µ 1 1, 0, 0 µ 2 1, 1, 0 Figure 5 Transition diagram for parallel system with M = 3 servers and U m,m 2 have the from given below: and W 0 = λ, U 0,3 = λ, D 1 = µ 3 µ 2, W 1 = µ µ 2 0, U 1,3 = 0 1, µ µ 1 D 2 = µ 2 µ 3 0 µ 1 0 µ 3, W 2 = µ 2 µ µ 1 µ 3 0, U 2,3 = 0 2, 0 µ 1 µ µ 1 µ 2 D 3 = [ µ 1 µ 2 µ 3 ], W 3 = (µ 1 + µ 2 + µ 3 ) The following rate matrices readily follow from Eq (25), or equivalently the first step of Algorithm 2 R 0 = D 1 (W 0 ) 1 = 1/λ µ 3 µ 2, R 1 = D 2 (W 1 ) 1 = µ 2/µ 3 µ 3 /µ 2 0 µ 1 /µ 3 0 µ 3 /µ 1 0 µ 1 /µ 2 µ 2 /µ 1 µ 1 R 2 = D 3 (W 2 ) 1 = [ µ 1 /(µ 2 + µ 3 ) µ 2 /(µ 1 + µ 3 ) µ 3 /(µ 1 + µ 2 ) ] Thus, we can find the steady state distribution using Eqs (27), (29), (26) or equivalently the remaining steps of Algorithm 2 Note that in this case T0 3 is a scalar and δ 3 = 1 T0 3 = m R m=0 k=0 2 k1 m = 1 + R 2 R 1 R R 2 R R 2 1 2,